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Discrete Dichotomies and Bifurcations from Critical Homoclinic Orbits

✍ Scribed by Flaviano Battelli; Claudio Lazzari

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
325 KB
Volume
219
Category
Article
ISSN
0022-247X

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✦ Synopsis

Perturbed discrete systems like x n+1 = f x n + ¡g x n ¡ , x n ∈ N , n ∈ , when the associated unperturbed map (¡ = 0) is not invertible and has a critical orbit γ n homoclinic to a hyperbolic fixed point p are studied. By critical we mean that the f γ n are invertible for any integer n = 0 but f γ 0 is not invertible. The main goal is to give sufficient conditions for a bifurcation from zero to many homoclinics when the parameter crosses zero. We also give a Melnikov like result assuring the persistence of homoclinics in a complete neighborhood of ¡ = 0. This result is similar to the ones obtained for diffeomorphisms and flows.

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